Over the past few years I have participated in a number of competitions with different degrees of difficulty. Here are some projects you may consider doing either individually or with friends and family. Keep in mind you can do this as a group project or a friendly competition.
Rubberband Propelled Plane:
Take a simple balsa rubberband propelled glider plane, which you can easily find at a hobby store (or order from online), and alter it to see how long you can keep it in the air. You can try changing the center of gravity or maybe how big the wing and/or tail is. Another aspect you can experiment with is the weight of different components. Do so by trial and error or, for those more numerically inclined, make some spreadsheets. Take it up a notch by building your own glider altogether!
Helpful hint: Soak the rubberband in soap water to increase its elasticity.
The Remote Control (RC) Plane:
Go to a hobby store and pick up a kit. These kits include all the necessary wood and blueprints for you to cut your parts out. There is also a step by step instruction manual if this is your first time. Adult supervision is advised for younger kids as this would require using razor blades or other sharp objects. Since this project is on a larger scale, include your family members or friends for extra pairs of hands.
After you have done so once, you can try out your own design by modifying the model kit you had bought/will buy. For the more ambitious, sketch your own design and build a plane from scratch. Hobby stores have lots of wood and the staff are quite knowledgeable when it comes to what kind of parts you should use, what type of propeller to use, or how stiff your control wires need to be.
Hopefully by the end you will have a moving plane like our UCSD DBF team: